As an undergraduate at Yale in the '70s I heard a variation on the basic legend, which I'll spell out a little since it has a slightly different moral from any others above.
Student goes to advisor saying I'd like to do a thesis generalizing the results in article X. Advisor (and I think I heard it with Milnor as the advisor) says, "I don't recommend that because I don't think that's a very good article." Student persists, writes thesis, states theorem at the defense and at that point the advisor rises to say "consider the following counterexample..."
I also heard a variation on "functions which turn out to be constant" legend. But the version I heard has the thesis getting accepted, the vacuity of it contents going unnoticed for several years until an undergraduate supplies a one-line proof.
John Myhill told me about junior faculty at the University of Chicago about to grade qualifying exams in their legendarily ruthless way. André Weil pops his head in the door and says "Pass them all, they're no worse than you are."